Game utilizing mathematical base systems

ABSTRACT

A game is disclosed having at least one playing board having an array of uniquely identifiable positions thereon with such positions having numerical value associated therewith. A position selection system is provided for sequentially identifying a plurality of said positions on the board and the numerical values associated therewith. When a preselected number of positions have been identified, further identification ceases; and a base selection card is chosen for selecting the particular mathematical base system on which continued play of the game is to be predicated. Once the base system is chosen, those identified numerical values which have mathematical significance in the particular base system chosen are converted to numerical values in another base system, preferably base ten, and the total of those converted numerical values are accumulated to represent the score of the player. A particular aspect of the invention resides in the disc-type converter which is of simple, compact construction and so designed to facilitate direct conversion of one numerical base system to another.

Tallaria States atent 1 GAME UTILIZING MATHEMATICAL BASE [21] Appl. No.:186,551

Related [1.8. Application Data [62] Division of Ser. No. 856,061, Sept.8, 1969, Pat. No.

[52] US. Cl. 235/88 [51] int. Cl G06c 3/00 [58] Field of Search 35/30,31 A, 31 C, 35/31 E; 235/85, 84, 88, 89 R, 70 R, 70 A, 61 B, 61 A, 61 R[56] References Cited UNITED STATES PATENTS 3,055,121 9/1962 Neal 35/31A 3,071,320 1/1963 Scott.... 235/61 A 3,332,156 7/1967 Reeves 35/303,352,031 11/1967 Lindquist 35/30 3,461,572 8/1969 Schmidt et a1. 35/31E 3,654,437 4/1972 Wyatt et a1. 235/84 3,654,438 4/1972 Wyatt et a1.235/84 [451 Nov. 6, 1973 Primary Examiner-Richard B. Wilkinson AssistantExaminerU. Weldon Attorney-Lawrence l. Lerner [57] ABSTRACT A game isdisclosedhaving at least one playing board having an array of uniquelyidentifiable positions thereon with such positions having numericalvalue as sociated therewith. A position selection system is provided forsequentially identifying a plurality of said positions on the board andthe numerical values associated therewith. When a preselected number ofpositions have been identified, further identification ceases; and abase selection card is chosen for selecting the particular mathematicalbase system on which continued play of the game is to be predicated.Once the base system is chosen, those identified numerical values whichhave mathematical significance in the particular base system chosen areconverted to numerical values in another base system, preferably baseten, and the total of those converted numerical values are accumulatedto represent the score of the player. A particular aspect of theinvention resides in the disc-type converter which is of simple, compactconstruction and so designed to facilitate direct conversion of onenumerical base system to another.

5 Claims, 6 Drawing Figures FAIENTEU 9 3 SHEET 1 UF 4 FIG. 1

:AIENTED 51975 3,770.192

SHEU 2 BF 4 BASE TEN L KEEP I SET l BONUS MODIFIED B I TOTAL SCOREPLAYER OR NUMERALS NUMERI ORE lNCLUDlNG PENALTY SEQUE E r 9g BONUS ORPENALTY l2' 6402 0402 En" I46 I46 12" m 4563 4503 5 I047 I047 12' 499603 0003 a 3 3 SET 2 BACKGROUND OF THE INVENTION This invention relatesto games which can be participated in by one or more players, and moreparticularly relates to such a game which is not only entertaining butwhich is also useful as a teaching device to instruct in themathematical principles underlying various concepts of the new math.

In the evolution of teaching mathematics, educators today tend to shyaway from characterising mathematics as a mental tool the rigors ofwhich must be arbitrarily accepted and memorized without understandingthe underlying principles thereof. For example, in the past, beginningstudents have been asked to accept the fact that the basic numberingsystem includes the digits 9 and that all higher numbers are based onvarious combinations of these digits which combinations are usuallymemorized without understanding the fundamental concepts of the basesystem upon which this numerical sequence is based. At the very most,these students subsequently learned to analyze a multi-digit number interms of a ones column, tens column, hundreds column, thousands column,etc., without appreciating the fact that each of the digits actuallyrepresents the particular digit times 10 to a prescribed exponent i.e.,10, 10, 10 10 etc. Today, not only are these underlying concepts of thebase ten system taught to beginning students, but also, these studentslearn that mathematical systems can employ other bases and that suchother base systems have utility in our own environment e.g., the basetwo system'as applied to computers. Hopefully, the end result is akeener understanding of the base 10 system, an awareness of theexistence and utility of other base systems, and an overall greaterappreciation for mathematics in general.

With such emphasis on the new math designers of educationallearning aidshave been searching for new and improved methods and devices forteaching the fundamental concepts of the various base systems upon whichmathematical computation may be predicated. An example of such aninstructional aid is represented by the Neal US. Pat. No. 3,055,l2lwhich discloses an educational device for converting numbers between onebase system and any other and which requires more knowledge on the partof the user than can be expected of anyone who needs the device. TheNeal device is perhaps typical of the prior art over which the instantinvention is intended to be an improvement, in the sense that Nealrepresents nothing more than a tool for converting between one basesystem and another and provides nothing else in the way of fun,excitement, or stimulation which would entice a student to either usethe Neal convertor or to. attempt to understand the fundamentalmathematical concepts upon which it is based.

SUMMARY OF THE INVENTION In contradistinction to the Neal convertor, andindee'cl'all suchprior art devicesiand/or learning aids, the instantinvention provides an enjoyable, exciting, competitive, but yet simplegame which can be played by people of all ages quite independently ofwhether or not such people are interested in learning or in factunderstand the fundamental mathematical principles upon which it isbased. At the same time, however, continued play of the game of theinstant invention stimulates those people who are interested to discoverthe underlying mathematical concepts involved.

As will be described in greater detail, the game of the instantinvention includes at least one playing board having an array ofuniquely identifiable positions thereon. Much like a Bingo" card, aplurality of such positions have a numerical value associated therewith.Position selection means, in the preferred form an ordinary deck ofplaying cards, is provided for sequentially identifying a plurality ofthe positions and for identifying the numerical values associatedtherewith. In the practice of the game, the playing cards aresequentially uncovered, and the players employ markers to cover thosepositions on their boards which have been identified by the individualplaying cards. Once any player has covered a row or column on his board,this portion of the play is halted; and, utilizing options which areavailable to them, the players select a particular row or column ontheir board with which they wish to continue the game.

After the players have selected and recorded their chosen sequence, baseselection means, preferably in the form of a deck of base cards each ofwhich identifies a particular base system, is employed to select aparticular numerical base system upon which continued play of the gamewill be predicated. Once a particular base has been selected, theplayers are directed to retain (and record) only those numerals in theirselected row or column which have mathematical significance in the baseselected, and to treat all other nonmeaningful numerals as a zero.Finally, the numbers of the thus modified sequence of each player areconverted from the particular base system which was selected to thenumerical values which such modified sequence represents in anotherpreselected base system, preferably base 10. These converted values areaccumulated and represent the participants score for that particular setof the game. I

As a particularly advantageous feature of the instant invention, adegree of strategy and excitement is built into the game by employingthe laws of probability and pre-arranging selected ones of theparticipants playing boards such that (1) a player must make a strategicdecision as to which type of playing board to employ and (2) given aparticular board, which row or column the participant is going to choosefor continued play.

As a further feature of the instant invention, a simple, compact, andnovel converter is provided for directly converting numbers between theparticular base system selected and the base ten system upon which thescore is predicated. Conversely, the converter may be used to directlyconvert numbers in the base ten to their equivalent in any other basesystem. Since the converter of the instant invention provides a directconversion between base systems it allows all participants to play thegame regardless of their understanding of the mathematical processesinvolved. However, as the description hereof unfolds, it will becomereadily apparent that continued play of the game will inherentlystimulate the practicipants to explore the mathematics upon which it isbased. I

Accordingly, it is an object of the instant invention to provide anexciting, enjoyable, competitive and stimulating game which can beemployed as an instructional device for teaching mathematical principlesof numerical base systems.

Another object of the instant invention is to provide such a game whichcan be enjoyably played by people of virtually all ages, regardless ofthe level of mathematical proficiency which they have achieved.

Still another object of the instant invention is to provide such a gamewhich includes at least one playing board having an array of uniquelyidentifiable positions thereon, a plurality of such positions havingnumerical values associated therewith; position selection means forsequentially identifying a plurality of said uniquely identifiablepositions and for identifying the numerical values associated therewith;base selection means for selecting a particular numerical base system onwhich continued play of the game is to be predicated; and conversionmeans for converting those identified numerical values which havemathematical significance in the particular numerical base systemselected by said base selection means to numerical values in apreselectecl other base system.

Still another object of the instant invention is to provide a converterfor converting between numerical base systems which converter hasindependent application if so desired, and in its preferred form, isparticularly adapted to use in the game of the instant invention.

Still another object of the instant invention is to provide such aconverter which includes a planar member having a plurality of arcuatelysegmented distinguishable areas disposed thereon about a vertical axisthereof, each of said areas being associated with a particular numericalbase system and being provided with indicia which equates numericalvalues in that particular base system with numerical values in anotherpreselected base system; and selective viewing means rotatably mountedwith respect to said vertical axis for viewing selected portions of saidareas.

Another object of the instant invention is to provide such a gamewherein a preselected number of base cards, having a preselected numberof numerical base systems associated therewith, are utilized in the playof said game so as to establish a predetermined probability of retainingmathematically significant numbers for use in subsequent portions of theplay.

Yet another object of the instant invention is to provide such a gamewherein the mathematically most significant positions of the rows andcolumns of at least one of the playing boards are provided withpreselected numerical values so as to predictably vary the value of saidboard.

Yet another object of the instant invention is to provide such a gamewherein the players can be appropriately compensated for the fact thatthey may be employing a preselectively weighted playing card.

Other objects of the instant invention and a better understandingthereof may be had by referring to the following specification anddrawings in which:

FIG. I is a plan view of most of the components of the game of theinstant invention as they would be layed out for play;

FIG. 2 is a plan view of three playing boards utilized in the game ofthe instant invention;

FIG. 3 is a plan view of four of the base cards utilized in the game ofthe instant invention;

FIG. 4 is a plan view of a portion of the converting wheel of theinstant invention;

FIG. 5 is an exploded perspective view of the convetting wheel of theinstant invention; and

FIG. 6 is a plan view of a portion of a score pad which may be employedin connection with the game of the instant invention.

Turning to the Figures and with particular reference to FIG. 1, there isillustrated most of the components of the game 10 of the instantinvention. Broadly speaking, and as will be described in greater detail,the game includes a plurality of playing boards 12; position selectionmeans 14, in the preferred form, preselected cards of an ordinary deckof regular playing cards; base selection means 16, in the preferredform, a plurality of cards each bearing an indication of the particularbase upon which continued play of the game is to be predicated; and aconverter designated 18 in FIG. 5 utilized to convert numbers betweenone base system and another. Additionally, a plurality of transparentmarkers designated 20 in FIG. 1 are employed to cover the playing boards12 in a manner to be described below.

Turning to FIG. 2, there is illustrated in detail three of the playingboards 12 which for case if identification have been individuallyidentified as 12' 12" and 12" respectively. Using the card 12 asexempliary, it will be seen that each of the cards comprises an array ofrows 22, 24, 26, 28 and columns, 30, 32, 34 and 36 so as to uniquelydefine in the illustrated case, sixteen positions such as 38 each ofwhich has associated therewith a numerical value ranging from onethrough nine. As can be seen in FIG. 2, each of the comumns 30, 32, 34and 36 is uniquely characterized, in the illustrated embodiment by thesymbols which are traditionally known as clubs, diamonds, hearts, andspades suits associated with an ordinary deck of playing cards. Thuscolumn 30 may be thought of as the clubs column; 32 as the diamondscolumn; 34 is the hearts column; and 36 is the spades column. Thus it ispossible to uniquely identify each of the sixteen positions 38established on the board 12, 12", and 12'41 with a deck ofordinaryplaying cards. For example, if one turned over the two of hearts, thatwould correspond to the position 44 of the card 12".

In fact, and as noted with respect to FIG. 1, the position selectionmeans 14 of the instant invention actually comprises a deck of ordinaryplaying cards (not shown in detail) from which have been withdrawn thetens, jacks, queens, and kings of all suits, it being understood duringthe playing of the game that an ace is valued as one.

Thus it may be appreciated that the first part of play of the game ofthe instant invention is much like the popular game Bingo wherein somemeans is provided to sequentially identify a plurality of positions, andthe individual participants search for, identify, and cover suchpositions if they are present on the boards with which they are playing.Similarly in the instant invention, each participant selects a boardsuch as 12, 12", or 12" and is supplied with a plurality of markers suchas 20 illustrated in FIG. 1. Next, the individual playing card, hecovers that position with one of the markers 20. Much like Bingo," thisportion of the play is continued (that is the playing cards aresequentially revealed) until one player has covered an entire row orcolumn of his card with the markers 20.

In the illustration of FIG. 2, and as illustrated by the phantom showing20 it has been assumed that playing cards of the position selectionmeans 114 have been sequentially uncovered to reveal 3 of clubs, the 4of clubs, the five of diamonds, the eight of diamonds, the two ofhearts, the four of hearts, the six of hearts, the three of spades, thesix of spades, and the nine of spade. At this point this portion of theplay would be terminated, since the player having card 12" will havecompletely covered the row designated 22.

Before going further into the description of the play and the othercomponents required therefor, it should be pointed out that the four byfour array illustrated on the playing boards 12 of FIGS. 1 and 2 hasbeen chosen primarily because of its compatibility with the playingcards of an ordinary deck of cards. That is the four columns 30, 32, 34,and 36 can be conveniently designated with the suits of an ordinary deckof cards. If desired, however, a larger array, such as five by five orsix by six, can be utilized so long as (1) there is unique identifyinginformation for each of the columns; (2) so long as the positionselection means is designed to permit the identification of the selectednumber of columns; and (3) so long as the converting wheel, to bedescribed in further detail, is appopriately modified to take into theaccount the extra columns. As an example, afive by five array could beused for the boards 12 and the familiar Bingo letters could be used toidentify each of the columns.

Assuming that play has stopped because the player utilizing the board 12has completed his row 34, the players must next select the particularrow or column upon which they wish to continue play. In the presentgame, the players have two options as to which row or column they wishto choose. The stratergy involved in selecting the options will beexplained in greater detail. In the first option each player may selectany row or column on his board which contains at least three markers 20,with the uncovered numeral in such a row or column being treated as azero. In the second option, each player may select the row or column ofhis board corresponding to the row or column which has been completed bythe participant who halted the play (again, treating any uncoverednumber in that row or column as zero). For example, in the assumed game,the player employing board 12' could select column 34 (because it hasthree markers 20), or using the second option he could select row 22(because it corresponds to the completed row of the participantemploying board 12").

For the sake of illustration, let it be assumed that the playeremploying playing board 12' selects column 34. He then records thenumeral sequence of this column (treating the uncovered 9 as zero) inthe block 45 of the score pad of FIG. 6 which block 45 is located underthe column marked numerals and in the row corresponding to his name (inthis case simply designated 12 for the participant). In FIG. 6 thenumeral sequence 6402 has been appropriately entered in the block 45 ofthe score pad.

For the participant employing board 12" let it be assumed that hechooses the numeral sequence of the completed row 22. He thereforerecords the numeral sequence 4563 in the block designated 47 of thescore pad of FIG. 6.

Finally, let it be assumed that the participant employing board 12"chooses column 36 and therefore records the numeral sequence thereof,9603 (recall, he must treat the uncovered 5 as a zero) in the blockdesignated 49 of the score pad of FIG. 6.

Each player having now determined and preferably recorded the numericalsequence which he will employ, the next step is to determine upon whichbase system, the remainder of play will be predicated. This isaccomplished by using the base selection means broadly designated 16 inFIG. 1 which in the preferred embodiment comprises a plurality of basecards such as those illustrated at 46, 48, 50 and 52 of FIG. 3. Asshown, each of these base cards comprises a card upon which isdesignated a particular base system, cards 46 through 52 designatingbase two, six, 10 and 10 respectively. It will also be noted that thecards 46 and 50 also direct the participant to keep those particularnumerals which have mathematical significance in the particular basesystem indicated by the respective card. For example, on card 46 whichdesignates that the game shall continue in the base two system; sinceonly zero and one have meaning in a base two system, the card directsthe participants to keep" only the 1s" in the mathematical sequence theyhave chosen. Similarly, on the base card 50, wherein the base six systemhas been selected, the card directs that the participants keep thenumerals l, 2, 3, 4 and 5, since these are the only numerals which havesignificance in a base six system. It will be appreciated that in thefamiliar base 10 system (for instance selected by the cards 48 and 52),all the numerals 1 through 9 are retained and therefore there is nospecific direction to keep only a prescribed group of numers. Withrespect to the cards 48 and 52, the designations -2,000 and +2,000" willbe subsequently explained.

It will be appreciated that once the players have selected their row orcolumn as explained above, only one base card of the base selectionmeans 16 is turned over to reveal on which base continued play of thegame is predicated. For the sake of illustration, let it be assumed thatat this point in the game the base card 50 has been revealed such thatthe game will be based on base six from this point on.

Once the card 50 has been overturned revealing the base six as thesystem, the participants review their previously selected rows orcolumns and modify the numerical sequence thereof by doing just as thebase card instructed, that is, keeping the numerals one through five anddiscarding (treating as zero) any of the numbers of their sequence whichhave no mathematical significance in the base six. For example, anddealing with the participant employing playing board 12, who haspreviously selected the numeral sequence 6402, his modified numericalsequence will become 0402 since the six has no mathematical significancein a base six system. This modified sequence 0402 has been appropriatelyplaced in the block 53 of the score pad of FIG. 6. Additionally thebasesix has been written into the small box 54 located beneath the columndesignated base.

For the participant employing playing board 12'', his numerical sequence4563 becomes a modified numerical sequence of 4503 since the 6 thereofhas no significance in base six system. Similarly for the participantemploying playing card 12" his numerical sequence 9603 becomes 0003since the nine and six thereof have no significance in a base sixsystem.

The final step in the game of the instant invention is to convert themodified numerical sequences of the individual participants to a totalscore. This is accomplished by converting the modified numericalsequence in the particular base selected to its numerical value in someother preselected base system. In the instant invention, the conversionis always to the base 10 system such that to determine the score of eachplayer it is necessary to convert their modified numerical sequences inthe base six to its equivalent value in the base 10. This isaccomplished by utilizing the converting wheel 18 of FIGS. 4 and hereof.

Without going into great detail at the present time as to the converter18, it will be pointed out that such converter includes a planar member56 which is divided into a plurality of arcuately segmenteddistinguishable areas disposed about a vertical axis 58 thereof. Each ofthese areas sequentially designated 60 through 68 in FIG. 4 correspondsto one particular base system, in this case base two through base sixrespectively; it being understood that on the undersurface of the planarmember 56 are the remaining segmented areas 70 through 74 correspondingto the base systems seven through nine.

Dealing with the segmented area 68 for example, each of theseareas'contains the necessary indicia for converting a four digit numberbetween the particular base system (base six) to numerical values in thebase system. Thus in the segmented area 68, there are four partial ringsof information 76, 78, 80 and 82 which contain the necessary equivalentsfor converting a four place number in the base six to the values ofthese digits in the base 10. For example, the number 4,444 in the basesix equals the sum of 864 144 24 4 1,036. In other words, 4,444 in the.base six equals 1 ,036 in the base ten. It should be noted that in theconverter of the instant invention, the pairs of numerals provided inthe partial rings ofinformation such as 76 to 82 are the fullequivalents in the respective mathematical base systems, there being nointermidiate step necessary in the conversion process (such asmultiplying 4 times 6 which is the fundamental mathematical operationinvolved in converting the third most signiticant digit in 4,444 to the144 equivalent in the base 10 system. Thus the converter table on themember 56 of FIG. 4 provides a direct, explicit conversion between onebase system and another.

Returning now to the hypothetical game, it can be seen from thesegmented area 68 of FIG. 4 that the modified numerical sequence 0402 ofthe participant employing playing board 12' would equal the sum of 0(from the conversion pair 84 of the partial ring 76); 144 (from theconversion pair 86 of the partial ring 78); zero (from the conversionpair 88 of the partial ring 80) and 2 (from the pair 90 of the partialring 82) for a total score of 146 recorded in the block 92 of the scorepad of FIG. 6. Similarly, the modified numoral sequence 4503 in the basesix for the participant employing playing board 12" would, from thesegmented area 68 of FIG. 4, equal the total of 864, 180, 0 and 3 or atotal score of 1,047. Finally the modified numerical sequence 0003 inthe base six for the participant employing playing board l2' would equala total score of 0 0 0 3 or 3. Thus, and at least for the present time,the winner of this particular set would be the participant employingplaying board 12" who had a total score of 1 ,047.

It is important to note throughout the above description that althoughthe conversion process involved transferring between two different basesystems; the participants, in using the conversion tables on the planarmember 56 of FIG. 4, did not have to be aware of the mathematicalprinciples therebehind. For example, it was not necessary for theparticipant of playing board 12' to understand why in the third mostsignigicant position 86 of partial ring 78, the number 4 in the base 6was equal to 144. That is he did not have to multiply 4 times 6 tounderstand either exponentsor bases to obtain the conversion. All he hadto understand was addition to get his aggragate score.

As noted previously, a certain degree of strategy and excitment is addedto the game of the instant invention by preselectively arranging certainof the playing boards 12 such that some of the boards are potentiallymore valuable than other of the boards. As a matter of fact, of thethree playing boards illustrated in FIG. 2, board 12' is what may bedesignated as an average board, board 12" is a good playing board, whileboard 12" is a poor playing board. This fact is fairly well representedby the scores achieved in the hypothetical game described above. Themanner of weighting the particular arrays will now be explained.

To begin with, it should be apparent that roughly speaking, the mostimportant numbers on the playing boards, so far as potential score isconcerned, are those numbers residing in rows 22 and columns 30. This ininherently so because the numbers in these rows and columns aremathematically most significant, i.e., they represent the fourth placeof a four digit number and accordingly, are always multiplied by somebase to the third power. Therefore, in order to make some of the playingboards 12 of greater or lesser value than others, it is only necessaryto concentrate on the distribution of numbers in the rows 22 and columns30 thereof.

To illustrate the system chosen, consider a deck of base cards (baseselection means 16) having twenty base cards of the type illustrated inFIG. 3; and that of these 20 base cards, there are two each calling forbases two through nine and four base cards calling for base 10. Oncehaving established the number and type of base cards it is possible todetermine the probability or retaining a particular digit as the basecards are uncovered. For example, the probability of retaining the digitone is 20 out of 20 since all bases from two through ten employ a l."Another way of saying this is that the probability of retaining a l is20/20 1. At the other extreme, the probalility of retaining a nine inthe game is only four out of 20 since only four of the 20 base cardscall for base 10 which is the only base system which has use for thenumber nine. Thus the probability of retaining a nine is 4/20 0.2

The probability for each of the numbers one through nine is set forthbelow as follows:

P (9) 4/20 =0.2 where P is the probability of retention of a particulardigit.

Having assigned a probability of retention to the individual numbers onethrough nine, it is then possible to establish an expectation value forthe digits one through nine by multiplying the probability of retainingthe individual digit times its value. For example, the expectation valuewhich can be assigned to the number two equals 0.9 X 2 1.8. Similarly,expectation values for the remaining digits are set forth below:

e (l) 1. X l l e(9)=0.2 9= 1.8 where e may be said to be the expectationvalue associated with the digits one through nine.

Knowing the expectation values of each of the digits, it is possible toassign a total expectation value for the playing boards 12 simply byadding the expectation values of each of the numbers in the rows andcolumns corresponding to 22 and 30 in FIG. 2. Without bothering to totalup the expectation value for each of the cards of FIG. 2, it can benoted from the expectation values listed above that as a rough estimate,those playing boards 12 whose rows 22 and columns 30 include the mostfives and sixs will have the greatest likelihood of producing a highscore in the play of the game (note that fives and si'xs have thehighest expectation value of 3.0). Thus by selectively distributingfives and sixs and the other numbers as well) in the first row and firstcolumn of the playing boards, it is possible to preselectively vary thepotential of a given playing board.

Of course, as the participants continue to play the game, they willbegin to realize that certain cardsproduce more winning scores thanothers. Eventually they may appreciate or they can be told that for aquick assessment a card that has the most fives and sixs in the firstrow and column thereof is potentially, based on the laws of probability,the better card with which to play. Alternatively, and in fact in thepreferred embodiment of the instant invention, weighted cards are soindicated by color coding. For example, a red dot 92 on the board 12" ofFIG. 2 indicates that the board 12" is a better board. Similarly, agreen dot 94 on the playing board 12" indicates that this board is apoor with which to play. Thus, the participants know before they choose,that particular boards are better or worse than others. Of course, atfirst blush it would appear that the participants would always choosethose boards having the red dot 92.

However in accordance with the invention, certain ones of the basecards, for instance 48 and 52 of FIG. 3, include instructionalinformation which will compensate the players for their choice of boardsshould that particular base card be uncovered during the play of thegame. For example, should the base card 48 be uncovered during the baseselection process; not only does that card indicate that play willcontinue in the base ten system, but the red dot 96 thereon also directsthat all players employing a red dotted playing board 12 must subtract2,000 from their score. For example, in FIG. 6 a 2000 has been placed inthe block 98 under the appropriate column entitled bonus or penalty andacross from the row designated 12" representing the fact that during thesecond set, base card 48 was selected such that the participantemploying board 12" was penalized. Similarly, should a base card such as52 be turned over during the base selection operation, the green dot 100thereon directs all players using a green dotted board such as 12 to add2,000 to their score. If desired, certain situation cards indicatingpenalty and bonus signals, but not naming a base, may be distributedthroughout the deck of base cards.

Thus when a participant initially makes his choice of playing boards 12,he has to make a decision as to whether (1) he will select a betterplaying board, in which case it is possible to be penalized if the cardsuch as 48 of FIG. 3 turns up in the base selection process; (2) tochoose a poor board such as 12", with the hope that a bonus base cardsuch as 52 of FIG. 3 comes up during the base selection process; or (3)whether to choose just an average card in which there is no possibilityof either penalty or bonus. This is a strategic decision on his part andwould depend to a great extent on his total score relative to the otherplayers as a new set gets under way, and also on which base cards havebeen uncovered in previous sets.

A second degree of strategy introduced by the laws of probability andexpectation values worked out for the instant invention has to do withthe options which are available to the players when they have to pick aparticular row or column upon which to continue the game. It will berecalled that the boards are covered (by turning the playing cards oneat a time) until some player covers a row or column of four numbers. Atthat time, every player has the option of choosing any row or column onhis playing board which has at least three markers thereon or choosingthe row or column of his board which corresponds to the completed row ofanother participants board.

Considering the play outlined before, it was assumed that participantemploying board 12" had stopped the play at this point by virtue ofhaving covered his row 22. At that time, it was assumed that theparticipant employing board 12" had selected column 36 (this waspermitted since it had three markers 20 thereon) with which to continueplay. Actually the participant employing board 12" had the option ofusing row 22 for two separate reasons: (1) this row also had threemarkers 20 thereon and (2) it additionally corresponded to the row 22covered by the participant employing board 12''. From previousdiscussion, it was pointed out that the expectation value of a nine (thefirst digit in column 36 of board 12") was 1.8 whereas the expectationvalue of a three (the first digit in row 22 of the board 12") is 2.4.Therefore, strategy would have dictated, at least during the first setof the game, that the player employing board 12 should have selected row22 rather than column 36. If he had done so, he would have achieved ahigher score (660 to be exact).

Preferably the game is practiced by laying a plurality of sets each oneidentical to the set which has been described thus far. If the playersare informed that there are only 20 base cards having two base cards foreach of the numbers two through nine and four cards for the base 10 (asin the example), he can to a certain extent gear his strategybyrecalling which base cards have been unturned during previous sets. Forexample, if the participant employing board 12" was aware that all ofthe base ten cards had been unturned during previous sets (and recallthat there are only four in the illustrated example), he most certainlywould not choose column 36 in the example given since he would know thatthere would be no possibility of using the nine which is the mostsignificant digit thereof. On the other hand, if no base ten card hadbeen unturned in previous sets, he might take a calculated gamble that abase ten card would come up during this particular base selectionoperation, with the hope of making a large score in that set.

It should be noted that if, in a particular set, a base 1 card such as48 or 52 of FIG. 3 is turned over during the base selection process,there is no need to use the conversion wheel of FIGS. 4 and 5 since bydefinition the four digit number utilized is already in the base 10.

Turning now to FIG. 5, and the details of the converter wheel 18, it waspreviously described with respect to FIG. 4, that the converter wheelincluded the planar surface 56 on opposite sides of which were disposedthe distinguishable segmented areas 60 through 74 corresponding to thebase systems two through nine with each one of such areas containing thenecessary indiciato directly convert the numbers utilized in therespective base system to their equivalent value in the base system.

For instance, the segment 68, drawn in detail to illustrate conversionfrom base six to base 10, includes the partial rings 76 through 82 eachbeing divided into six pairs of numbers necessary to make the directconversion required from a four digit number in the base six to valuesin the base 10. Likewise, the segment designated 64 in FIG. 4 containsfour partial rings of information each ring having two pairs of numbersnecessary to convert a'four digit number in the base two systern to theequivalent in the base 10 system. It should be noted that for the sakeof drawing simplicity, the detailed indicia of the segments 62 through66, and the total undersurface of the planar surface 56 has beeneliminated.

Turning to FIG. 5, and dealing only with the upper half thereof (itbeing understood that the lower half is a identicallto facilitate theconversion process, there are provided a plurality of disc-like members102, 104, I06 and 108 each of which is individually rotatably mountedabout the axis 58, for example by means of a threaded pin 109 beinginserted through the entire assembly along the axis 58. The disc'likemember 102 includes a viewing aperture 110 which is radially spaced fromthe axis 58 so as to overly the ring of information defined by thepartial rings 82 disposed on the planar surface 56 therebeneath.Similarly the disc-like memher 104 includes a viewing aperture 112radially disposed from the axis 58 so as to overly the ring ofinformation defined by the partial rings 80. In like manner, thedisc-like member 106 includes a viewing aperture 114 radially spacedfrom the axis 58 so as to overly the ring defined by the partial rings78, and the disc-like member 108 includes a viewing aperture 116radially spaced from the axis 58 so as to overly the ring of informationdefined by the partial rings 76 on the planar surface 56.'Furthermore,in order to make sure that the lower disc-like members do not block theviewing apertures of the disc-like members thereabove, lower disclilcemembers 104, 106 and 103 include arcuate cut out sections 118, I20 and122 respectively.

In operation, and working with the previous example, let it be assumedthat it is desirable to convert 0402 (the modified numerical sequence ofthe participant employing board 12') to its value in the base 10 system.The user simply rotates the disc-like members 102 through 108 such thatthe viewing aperture 116 overlies the position identified as 84 in FIG.4; the viewing aperture 114 overlies the position 86 in FIG. 4; theviewing aperture 112 overlies the position 88 of FIG. 4; and the viewingaperture overlies the position 90 of FIG. 4. In this manner, the usercan simply read off the equivalant values in the base 10 and need onlyadd the individual values to achieve his score of 146. As notedpreviously, the convertor wheel 18 provides the necessary conversiondirectly, i.e., each pair gives the equivalant numerical values, suchthat the user need not understand the mathematical concepts involved toutilize the converter.

Finally when a converter employing the above described inventiveconcepts, is specificially designed for use in the game of theinstantinvention, it can be tailored to facilitate play. Thus, and withreference to FIG. 4, along the outer periphery of the segmented areas ofthe planar surface 54, there are provided distinguishable color codedarcuate segments 124, 126, I28, and 132 which by pre-design have thesame colors as the ink utilized with respect to base numbers on the basesection cards of FIG. 3. For example, for the base card 50 of FIG. 3,the base six would be printed in blue ink, and similarly the arcuatesegment 132 of FIG. 4 would be coded with a blue coloring. In thismanner, when a base six card comes up the-user quickly knows to rotateall of the disc-like members 102, 104, 106 and 108 to that section ofthe planar surface 56 which is defined by the blue arcuate segment 132imprinted thereon before he individually rotates the disc to find outhis modified sequence.

Additionally, and with respect to the converter per se, those numbers ofthe pairs in the segment 68 which represents base six values are coloredblue also. This helps the user to identify which numeral of each pairrelates to base six, the remaining (black) numeral being the equivalentin base ten.

Although the converter 18 of FIG. 5 is illustrated in explodedperspective view, it will be appreciated that in its assembled conditionall discs are closely stacked one upon another such that the overallthickness of the converter may be in the order of a quarter inch. Thiscompact, easy to use converter wheel is simple to manufacture, simple toassemble and therefore in of itself presents a significant improvementover similar converter wheels presently available.

It will be appreciated that the converter 18 of FIG. 5 can be used forthe converse operation, i.e., to determine the equivalent of a base IDnumber in any other base. As will be shown, by virtue of the instantinvention, this conversion can be performed with relative ease employingonly addition and subtraction and does not require multiplication,division or a knowledge of exponents.

For example, suppose it is desired to convert 128 in the base ten systemto its equivalent number in the base six system. One simply rotates thedisc-type members 102 through 108 over the base six area 68 until hefinds the largest numeral less than 128. It can be seen in FIG. 4, thatthis would be 108 which would appear adjacent the numeral 3 through theviewing aperture 114 of disc-like member 106. The difference between 128and 108 is 20. Therefore, one next examines the numbers in area 68 andfinds the largest number less than 20. This is the number 18 of the pair3, l8 and would appear through the viewing aperture 112 'of the disclike member 104. Next one takes the difference between 20 and 18 whichis 2. He would find on area 68 that the equivalant of 2 in the base 10is 2 in the base 6. This pair of numbers 2, 2 would be seen when theviewing aperture 110 of disc-like member 102 was rotated over theportion 90 indicated in FIG. 4. Thus; 128 332 or in other words, 128 inthe base 10 may be expressed as 332 in the base six.

From'the above it will be seen that the entire game of the instantinvention can be played without the participants understanding theunderlying mathematical concepts. However, it should be apparent thatcontinued play will stimulate the minds of the participants intoattempting to apply the laws of probability and additionally shouldstimulate some participants to inquire further into the fundamentals ofthe various base systems. The additional information necessary forunderstanding the base systems and/or the laws of probability can beprovided in an appropriate portion of the instruction booklet whichaccompanies the game and/or can be provided by an instructor.

Although this invention has been described with respect to its preferredembodiments it should be understood that many variations andmodifications will now be obvious to those skilled in the art, and it ispreferred, therefore, that the scope of the invention be limited, not bythe specific disclosure herein, only by the appended claims.

' lclaim:

l. A converter comprising:

a planar member having a plurality of arcuately segmenteddistinguishable areas disposed thereon about a vertical axis thereof;

each one of said areas being associated with a particular numerical basesystem and being provided with indicia which equates numerical values inthat particular base system with the numerical value of said numericalvalues in another preselected base system;

selective viewing means rotatably mounted with respect to said verticalaxis for viewing selected portions of the indicia of said area; andwherein the indicia in each of said areas includes a plurality ofpartial rings of information corresponding in number to the desirednumber of digits it is desirable to convert from one base system to saidother preselected base system.

2. The converter of claim 1 wherein said planer member is provided withidentifying information for identifying the numerical base systemassociated with each area.

3. The converter of claim 1 wherein each of said partial rings ofinformation includes a plurality of pairs of numbers showing theequivalence of a numerical value in the respective base system to anumerical value in said other preselected base systems; the number ofpairs of numbers in each partial ring being equal to the particular basesystem to which the respective area corresponds.

4. The converter of claim 3 wherein said selective viewing meanscomprises a plurality of disc-like members stacked one above the otheron said planar member, each of said disc-like members having an apertureradially located from said axis to rotatively overly a respective one ofthe total rings of information defined by the sum of said partial ringsof information each of said apertures being dimensioned to reveal one ofsaid pairs of numbers.

5. The converter of claim 4 wherein successive lower ones of saiddisc-like members include annular removed portions so as tonon-interruptingly permit pairs of numbers on said planar members to beviewed by the aperture in successively higher ones of said disc-likemembers.

* a a a

1. A converter comprising: a planar member having a plurality ofarcuately segmented distinguishable areas disposed thereon about avertical axis thereof; each one of said areas being associated with aparticular numerical base system and being provided with indicia whichequates numerical values in that particular base system with thenumerical value of said numerical values in another preselected basesystem; selective viewing means rotatably mounted with respect to saidvertical axis for viewing selected portions of the indicia of said area;and wherein the indicia in each of said areas includes a plurality ofpartial rings of information corresponding in number to the desirednumber of digits it is desirable to convert from one base system to saidother preselected base system.
 2. The converter of claim 1 wherein saidplaner member is provided with identifying information for identifyingthe numerical base system associated with each area.
 3. The converter ofclaim 1 wherein each of said partial rings of information includes aplurality of pairs of numbers showing the equivalence of a numericalvalue in the respective base system to a numerical value in said otherpreselected base systems; the number of pairs of numbers in each partialring being equal to the particular base system to which the respectivearea corresponds.
 4. The converter of claim 3 wherein said selectiveviewing means comprises a pluraLity of disc-like members stacked oneabove the other on said planar member, each of said disc-like membershaving an aperture radially located from said axis to rotatively overlya respective one of the total rings of information defined by the sum ofsaid partial rings of information each of said apertures beingdimensioned to reveal one of said pairs of numbers.
 5. The converter ofclaim 4 wherein successive lower ones of said disc-like members includeannular removed portions so as to non-interruptingly permit pairs ofnumbers on said planar members to be viewed by the aperture insuccessively higher ones of said disc-like members.